We study randomized stationary methods for symmetric positive definite linear systems in which component $j$ is selected with probability proportional to $|r_j|^\ell$. This power-weighted family interpolates continuously between uniform randomized Jacobi as $\ell \to 0$ and Gauss--Southwell greedy relaxation as $\ell \to \infty$. For the central case $\ell = 2$, we sharpen the standard one-step convergence analysis using the inverse participation ratio (IPR) $ν^2(r) = n\|r\|_4^4/\|r\|_2^4$, which equals $1$ when the residual is uniform and grows toward $n$ as it concentrates. The resulting bound amplifies the expected per-step progress by exactly $ν^2$ over the uniform-sampling baseline. The IPR can be computed online at $O(n)$ cost and doubles as a per-iteration diagnostic. We extend the analysis to asynchronous power-weighted Jacobi via the Avron--Druinsky--Gupta framework, obtaining an epoch-based convergence theorem in which the IPR controls both the progress coefficient and the allowed-delay window. Numerical experiments on shared-memory hardware support the sharpened bound and show the IPR trajectory is essentially concurrency-insensitive. Unexpectedly, consistent-reads execution, the easier case for the ADG analysis, destabilizes power-weighted sampling at high concurrency while inconsistent reads remain stable; the same IPR that amplifies progress amplifies a thread-collision rate that inconsistent reads appear to absorb. We propose a feedback-damping mechanism and verify two predictions about its dependence on problem size.

翻译：我们研究对称正定线性系统的随机平稳方法，其中分量 $j$ 以与 $|r_j|^\ell$ 成正比的概率被选中。这个幂权重族在均匀随机Jacobi（当 $\ell \to 0$ 时）和Gauss–Southwell贪婪松弛（当 $\ell \to \infty$ 时）之间连续插值。对于核心情形 $\ell = 2$，我们利用逆参与比（IPR）$\nu^2(r) = n\|r\|_4^4/\|r\|_2^4$ 来锐化标准的单步收敛分析，当残差均匀时该比值为1，当残差集中时向 $n$ 增长。所得边界恰好将均匀采样基线下的预期每步进展放大了 $\nu^2$ 倍。IPR 可以 $O(n)$ 代价在线计算，并兼作每次迭代的诊断指标。我们通过Avron–Druinsky–Gupta框架将分析扩展到异步幂权重Jacobi，得到了一个基于轮次的收敛定理，其中IPR同时控制进展系数和允许延迟窗口。在共享内存硬件上的数值实验支持了锐化后的边界，并显示IPR轨迹本质上不受并发性影响。出乎意料的是，对于ADG分析中更简单的一致性读取执行，在高并发下会破坏幂权重采样的稳定性，而非一致性读取则保持稳定；同一IPR在放大进展的同时也放大了线程碰撞率，而非一致性读取似乎能吸收这一碰撞率。我们提出了一种反馈阻尼机制，并验证了关于其与问题规模依赖关系的两个预测。